← Back to Problems
Mathematical Logic / Set TheoryResearchAI-Generated

Is it consistent with ZF set theory without the Axiom of Choice that every ultrafilter on the natural numbers fails to be a P-point, while also failing to satisfy any of the standard chain condition properties used in forcing arguments?

Related: Rudin-Keisler ordering of ultrafilters, Martin's Axiom and the countable chain condition, Shelah's model with no P-points

The problem asks whether we can construct a model of set theory without the Axiom of Choice in which ultrafilters on the natural numbers exist but none of them are P-points, and simultaneously the standard chain condition tools used in forcing, which normally rely on Choice, break down in ways that prevent us from controlling which ultrafilters survive. This sits at the intersection of two active research threads: the recent simplified proof that P-points may be absent, and the emerging study of how chain conditions behave in choiceless set theory. We want to understand whether these two phenomena can coexist in a single coherent mathematical universe, and what constraints one places on the other.

View Source Paper →